Fig. 1. Simplified scheme of SHG in a leaky cavity. FF (SH) beam is represented in red (blue), κ_in (κ_out) denotes the input (output) coupling coefficient and ζ the spatial overlap between the dominant modes at the two frequencies.

Fig. 2. Comparison between isolated (top) and arrayed (bottom) air-suspended GaAs nanoantennas with radius r = 200 nm and height h=400 nm. A normally impinging beam is linearly polarized along x, and the array (bottom) has a period of p=1 μm. (a) Scattering cross section decomposed in electric (blue) and magnetic (red) multipolar contributions, computed by projecting the field inside a nanoparticle onto vector spherical harmonics⁹⁵. The scattering cross section from the isolated particle has been scaled by a factor 6 to be compared with the plot on the bottom. The black dashed curve (bottom) displays the array transmission. (b) Near-field enhancement corresponding to the magnetic-dipole resonance.

Fig. 3. Arrays of quasi-independent dielectric χ⁽²⁾ Mie-type resonators. (a) SHG from a GaAs metasurface radiated into different diffraction orders vs. pump polarization angle. Green: total; blue: (0,±1) orders; red: (±1,0) orders; black: (0,0) order⁹⁸. (b) SH back-focal plane (BFP) imaging of a LiNbO₃ metasurface for horizontally (center) or vertically (right) polarized pump¹⁰¹. (c) Mechanism to partially redirect SHG from an AlGaAs metasurface close to normal direction⁹⁹. (d) Controlling SH polarization. BFP imaging of SH from isolated (left) or arrayed (right) AlGaAs nanoantennas with elliptical basis¹⁰⁰. (e) Broadband frequency generation in a GaAs metasurface through multiple nonlinear processes⁹⁶.

Fig. 4. Nonlinear wavefront shaping in dielectric metasurfaces.(a) Si metasurface implementing the first Kerker condition and funneling THG into the (-1)-diffraction order¹⁰⁶. (b) Top: TH focusing from a Si metasurface. Bottom: Nonlinear imaging and spatial correlation for two apertures¹⁰⁷. (c) SH beam-steering (left) and focusing (right) from an AlGaAs metasurface¹⁰⁸. (d) SH geometric phase control in a Si metasurface¹⁰⁹.

Fig. 5. Quasi-BIC mode generation in nonlinear metasurfaces. A few ways to break the symmetry and reveal the high-Q resonance are sketched in the top row. Yellow (blue) color denotes a material addition (removal). (a) L-shaped GaAs/AlGaO heterostructures¹²⁴. (b) Asymmetric Si nanodimers made of two rods with different widths¹²⁵. (c) T-shaped Si resonators¹²⁶. (d) Asymmetric GaP nanodimers made of two elliptical-basis cylinders rotated by an angle θ¹²⁷.

Fig. 6. SHG efficiency vs. Q-factor of the dominant FF mode for different resonant devices (markers’ color) and materials (markers’ shape).

Table 1. Most common materials for χ⁽²⁾ nonlinear nano-optics. Refractive index n and second-order nonlinear coefficient d all refer to a wavelength λ = 1550 nm but for LiNbO₃ for which we consider λ = 1313 nm.